This is a lot easier to do with LVM on top of raid. There, you would just install the new RAID1 devices, add them to your volume group, and issue a pvmove to have the extents moved off the old devices onto the new one(s).
Assuming that you don't start using LVM, about all that you can do is (depending if you have spare drive space) install the new two 250GB drives, create new raid partitions on them, mount them somewhere, and cp the old partitions onto the new ones. Change /etc/fstab on the new version of your root partition to use the new raid devices (instead of md0, md1, and md2). Add a new image section in your old lilo.conf file that will allow you to boot from the new /root partition device, re-run lilo and try to boot off the new partitions. Make sure that everything still works and you should be able to remove the old drives.
You can also just install one of the 250GB drives, create the RAID1 partitions with only one of the two drives present, do the same as above, and then add the other 250GB drive, partition it, and add the missing halves to your new RAID1 arrays. If you have the right type of enclosure, you can follow the directions in section 4.2.2 of /usr/doc/Linux-HOWTOs/Software-RAID-HOWTO to hot swap the drives.
I've got one of these
on my main computer. The top two drives are a RAID1 array for linux and the bottom drive is for my Windows "stuff". I've used the procedure in Software-RAID-HOWTO to remove one of the drives from the running raid array, disconnect it from the system, pull that drive out, replace it with the new one, attach the new one to the system, create my raid partition(s) on it, copy the old data across (via pvmove, which you can't use), and try a few test boots. When things were working to my satisfaction, I powered down the remaining old drive and swapped it out for the new one, adding the new partitions and assigning them to the appropriate RAID1 devices.
As long as you don't copy things the wrong way, you should have no problem. If you think that you may do that and don't have a proper backup, just pull one of the two RAID1 drives and leave it off your system until you are sure things are working correctly. Sort of an instant backup since you should be able to use that one to re-create your system.